When can eigenvalue equations be used?

[metroplex021]

Does anybody know what physical and mathematical conditions must be in place if we are to represent the states of physical systems using eigenvalue equations (so that we can represent the energy of a state |n> of a particle by H|n>=E|n>, etc?) Thanks!

 

[Fredrik]

|n> is the mathematical representation of the class of preparation procedures that ensure that the probability of getting the result E_n when you measure the system's energy is 1.

 

[A. Neumaier]

Does anybody know what physical and mathematical conditions must be in place if we are to represent the states of physical systems using eigenvalue equations (so that we can represent the energy of a state |n> of a particle by H|n>=E|n>, etc?) Thanks!

The |n> must be eigenstates of H in order that the equation H|n>=E_n|n> is valid.

 

[eaglelake]

Does anybody know what physical and mathematical conditions must be in place if we are to represent the states of physical systems using eigenvalue equations (so that we can represent the energy of a state |n> of a particle by H|n>=E|n>, etc?) Thanks!

Quantum mechanics answers the questions: If we measure the observable $$\hat A$$ for a specified experimental configuration, 1) what are the possible results of a measurement? And 2) what is the probability of obtaining each result? Answer 1) The possible results of a measurement are eigenvalues of $$\hat A$$. Answer 2) The probability of obtaining each result is $$\left| {\left\langle {{a_k }} \mathrel{\left | {\vphantom {{a_k } \psi }} \right. \kern-\nulldelimiterspace} {\psi } \right\rangle } \right|^2$$, where the $$\left| {a_k } \right\rangle$$ are the eigenvectors of $$\hat A$$. $$\left| \psi \right\rangle$$ is the state vector, which is determined by the experimental configuration. Thus, we need to know the eigenvalues and eigenvectors, which are obtained by solving the eigenvalue equation of the measured observable. Solving eigenvalue equations is part of doing quantum mechanics. If we are going to measure the energy then we must solve the energy eigenvalue equation $$H\left| n \right\rangle = E_n \left| n \right\rangle$$, where $$\left| n \right\rangle$$
is the eigenvector corresponding to the eigenvalue $$E_n$$. $$H$$ is the Hamiltonian operator.

If the state vector is $$\left| n \right\rangle$$, an eigenvector of the Hamiltonian, then a measurement of the energy always yields the value $$E_n$$. There is no uncertainty in energy when the particle is in an energy eigenstate. Generally, repeated measurements yield the entire eigenvalue spectrum. By blocking out all results except $$E_n$$, the particle will be in eigenstate $$\left| n \right\rangle$$.
Best wishes.

 

[metroplex021]

Thanks very much for these clarifications about the situation in QM, but what I'm wondering is a bit more general. What I'm wondering is whether it is a peculiarity of quantum mechanics that we can represent the values that physical systems have for various properties this way, or rather whether it's something that we can expect to be true of physical systems represented in theories quite generally. Is it the case, for example, that if we wanted we could represent the angular momentum or energy of a classical system using eigenvalue equations (even if we don't usually write them that way)? Or may we expect that, if X is a scalar-valued observable quantity of a system a that obeys an as-yet unknown theory that supplants quantum mechanics, then we should be able to write Xa =xa for some scalar x?

Basically what I'm wondering is whether the applicability of eigenvalue equations in physics is a peculiarity of QM, or a feature that we should expect to crop up in theories of physics quite generally. Any thoughts (or references) would be much appreciated!

 

[A. Neumaier]

whether it is a peculiarity of quantum mechanics that we can represent the values that physical systems have for various properties this way,

It is the essence of quantum mechanics. The established view is that all observables are represented by operators whose eigenvalues define their possible measurement values,
which are predictable with certainty if the system is in the corresponding eigenstate.

For example, the eigenvectors of the z-component J_3 of the angular momentum operator
give the states where the measurement of J-1 has a definite value.

This has no analogue in classical mechanics. It surely will survive coming changes in our theories, at least in a good approximation.